So we've been having issues lately with users not wanting to fill out the request type any further than the first pull-down, and having tickets lost in the ether because they aren't being routed to the right people.
Is there a way to make it so they can't submit the ticket without choosing ALL the categories when creating a new ticket?
Hmm... that is a bit odd. The request Type field is by default a required field and the ticket will not be saved if the client will not select all throughout the sub-request types. Maybe a few more info would help like which version are you running, was it just the clients or also techs are doing this or maybe you could be using another webform that sends the ticket through REST API.